Search Loci: Convergence:
The cowboys have a way of trussing up a steer or a pugnacious bronco which fixes the brute so that it can neither move nor think. This is the hog-tie, and it is what Euclid did to geometry.
In R Crayshaw-Williams The Search For Truth, p. 191.
Can You Really Derive Conic Formulae from a Cone?
Deriving the Symptoms of the Obtuse-angled Cone
Next we consider the obtuse-angled cone with vertex O and a plane intersecting a generating line OB at a right angle at point A. The plane intersects the cone in the amblytome with diameter AG.
A dynamic view of this construction will be helpful in what follows.
Now consider the similar triangles NAB and NCG in the axial plane plane through O, B, and C. The triangles are similar because they each have a right angle and opposite vertical angles. This in turn implies
Combining (1) and (2) we have
In the axial plane we also have similar triangles GCN and FDA, because they each have a right angle and they have another equal angle since AD is parallel to NC. (See Elements Book I, Prop. 29, a very famous proposition indeed. Follow the link to see why.) These similar triangles give us:
Finally, in the same axial plane are the similar triangles A'NC and A'AD. The triangles are similar because AD is parallel to NC in triangle A'NC (see for example, Elements Book VI, Prop. 2). From this we have
using the fact that AF = 2AL.
Just as in the last section, we can make this look more like the modern equation of a hyperbola by substituting x = AN, y = NP, 2a = AA', and p = 2AL. This gives us
which can be rewritten as the modern form of a hyperbola. (Do you see how?)